Mathematics 143
Set Theory - Terms and Symbols
I. Set notations
 (a) Capital letters : A, B, C, S, T (b) Curly brackets containing a list of elements. Example : {a, b, c} is the set containing the letters a, b, and c. (c) Set-builder notation (see Rolf Pg. 425). Example : {x | x satisfies property P} where P is any property which makes sense when applied to x. It may be either easy or difficult to determine whether or not property P is true for certain values of x. If the determination takes a very long time, such as a billion years, the set may have uncertain status.
II. Special sets
 (a) U : The UNIVERSE SET, i.e., the set of all things (or elements) under discussion at the moment. U changes from one problem to another. (b) Ø or { } : the EMPTY SET or NULL SET, containing no elements. Note that { } is different from the number "0" and the sets { 0 } and { Ø }. A common way to write the number "zero" is to slash it, so that it differs from the letter "O", but this practice must be avoided when the empty set enters a discussion.
III. Relations
(a)
 The following 6 statements all mean the same thing : "A B" , "B A" , "A is contained in B", "B contains A", "A is a subset of B", and "every element of A is an element of B".
(b)
(c) or =     "A B" and "A = B" mean that the sets A and B have precisely the same elements (as well as the same size).
(d) /     means "not" when drawn across any symbol representing a verb. For example, A B means "A is not the same set as B"
(e) e or indicates set membership. Thus, "x e A" means that "x is a member of the set A" or "x is an element of the set A".
IV. Operations
 (a) A B = {x | x e A   AND   x e B} (See Rolf Pg 428 Fig 6-4) (b) A B = {x | EITHER x e A   OR   x e B} (See Rolf Pg 427 Fig 6-3) (c) A' or Ac A' = {x | x is not in the set A} (See Rolf Pg 429 Fig 6-6) (d) n(A) or |A| n(A) = is the size of A, or the number of elements in A.
 V. Venn Diagrams Venn Diagrams are stylized pictures of sets. See exercises